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arXiv:2509.02689

Quantum DPLL and Generalized Constraints in Iterative Quantum Algorithms

Lucas T. Brady, Stuart Hadfield

Published Sep 2, 2025• 0 citations
arXivHTMLPDF

Abstract

Too often, quantum computer scientists seek to create new algorithms entirely fresh from new cloth when there are extensive and optimized classical algorithms that can be generalized wholesale. At the same time, one may seek to maintain classical advantages of performance and runtime bounds, while enabling potential quantum improvement. Hybrid quantum algorithms tap into this potential, and here we explore a class of hybrid quantum algorithms called Iterative Quantum Algorithms (IQA) that are closely related to classical greedy or local search algorithms, employing a structure where the quantum computer provides information that leads to a simplified problem for future iterations. Specifically, we extend these algorithms beyond past results that considered primarily quadratic problems to arbitrary k-local Hamiltonians, proposing a general framework that incorporates logical inference in a fundamental way. As an application we develop a hybrid quantum version of the well-known classical Davis-Putnam-Logemann-Loveland (DPLL) algorithm for satisfiability problems, which embeds IQAs within a complete backtracking based tree search framework. Our results also provide a general framework for handling problems with hard constraints in IQAs. We further show limiting cases of the algorithms where they reduce to classical algorithms, and provide evidence for regimes of quantum improvement.

DOI
10.48550/arxiv.2509.02689

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Too often, quantum computer scientists seek to create new algorithms entirely fresh from new cloth when there are extensive and optimized classical algorithms that can be generalized wholesale. At the same time, one may seek to maintain classical advantages of performance and runtime bounds, while enabling potential quantum improvement. Hybrid quantum algorithms tap into this potential, and here we explore a class of hybrid quantum algorithms called Iterative Quantum Algorithms (IQA) that are closely related to classical greedy or local search algorithms, employing a structure where the quantum computer provides information that leads to a simplified problem for future iterations. Specifically, we extend these algorithms beyond past results that considered primarily quadratic problems to arbitrary k-local Hamiltonians, proposing a general framework that incorporates logical inference in a fundamental way. As an application we develop a hybrid quantum version of the well-known classical Davis-Putnam-Logemann-Loveland (DPLL) algorithm for satisfiability problems, which embeds IQAs within a complete backtracking based tree search framework. Our results also provide a general framework for handling problems with hard constraints in IQAs. We further show limiting cases of the algorithms where they reduce to classical algorithms, and provide evidence for regimes of quantum improvement.
One of the challenges of quantum computers in the near- and mid- term is the limited number of qubits we can use for computations. Finding methods that achieve useful quantum improvements under size limitations is thus a key question in the field. In this vein, it was recently shown that a hybrid classical-quantum method can help provide polynomial speed-ups to classical divide-and-conquer algorithms, even when only given access to a quantum computer much smaller than the problem itself. In this work, we study the hybrid divide-and-conquer method in the context of tree search algorithms, and extend it by including quantum backtracking, which allows better results than previous Grover-based methods. Further, we provide general criteria for polynomial speed-ups in the tree search context, and provide a number of examples where polynomial speed ups, using arbitrarily smaller quantum computers, can be obtained. We provide conditions for speedups for the well known algorithm of DPLL, and we prove threshold-free speed-ups for the PPSZ algorithm (the core of the fastest exact Boolean satisfiability solver) for well-behaved classes of formulas. We also provide a simple example where speed-ups can be obtained in an algorithm-independent fashion, under certain well-studied complexity-theoretical assumptions. Finally, we briefly discuss the fundamental limitations of hybrid methods in providing speed-ups for larger problems.
Quantum algorithms have been widely studied in the context of combinatorial optimization problems. While this endeavor can often analytically and practically achieve quadratic speedups, theoretical and numeric studies remain limited, especially compared to the study of classical algorithms. We propose and study a new class of hybrid approaches to quantum optimization, termed Iterative Quantum Algorithms, which in particular generalizes the Recursive Quantum Approximate Optimization Algorithm. This paradigm can incorporate hard problem constraints, which we demonstrate by considering the Maximum Independent Set (MIS) problem. We show that, for QAOA with depth p=1p=1p=1, this algorithm performs exactly the same operations and selections as the classical greedy algorithm for MIS. We then turn to deeper p>1p>1p>1 circuits and other ways to modify the quantum algorithm that can no longer be easily mimicked by classical algorithms, and empirically confirm improved performance. Our work demonstrates the practical importance of incorporating proven classical techniques into more effective hybrid quantum-classical algorithms.
Quantum algorithms have been widely studied in the context of combinatorial optimization problems. While this endeavor can often analytically and practically achieve quadratic speedups, theoretical and numeric studies remain limited, especially compared to the study of classical algorithms. We propose and study an alternative class of hybrid approaches to quantum optimization, termed iterative quantum algorithms, which in particular generalizes the recursive quantum approximate optimization algorithm (QAOA). This paradigm can incorporate hard problem constraints, which we demonstrate by considering the maximum independent set (MIS) problem. We show that, for QAOA with p=1p=1p=1 circuit layers, this algorithm performs exactly the same operations and selections as the classical greedy algorithm for MIS. We then turn to deeper p>1p>1p>1 circuits and other ways to modify the quantum algorithm that can no longer be easily mimicked by classical algorithms, and empirically confirm improved performance. Our work demonstrates the practical importance of incorporating proven classical techniques into more effective hybrid quantum-classical algorithms.
While the development of quantum computers promises a myriad of advantages over their classical counterparts, care must be taken when designing algorithms that substitute a classical technique with a potentially advantageous quantum method. The probabilistic nature of many quantum algorithms may result in new behavior that could negatively impact the performance of the larger algorithm. The purpose of this work is to preserve the advantages of applying quantum search methods for generalized pattern search algorithms (GPSs) without violating the convergence criteria. It is well known that quantum search methods are able to reduce the expected number of oracle calls needed for finding the solution to a search problem from O(N) to O(N) However, the number of oracle calls needed to determine that no solution exists with certainty is exceedingly high and potentially infinite. In the case of GPS, this is a significant problem since overlooking a solution during an iteration will violate a needed assumption for convergence. Here, we overcome this problem by introducing the quantum improved point search (QIPS), a classical–quantum hybrid variant of the quantum search algorithm QSearch. QIPS retains the O(N) oracle query complexity of QSearch when a solution exists. However, it is able to determine when no solution exists, with certainty, using only O(N) oracle calls.
We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.
$ $ We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on nnn variables, or outputs that none exists, and whose corresponding tree contains TTT vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(Tn3/2log⁡n)O(\sqrt{T} n^{3/2} \log n)O(T​n3/2logn) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree.